Solve the Poisson equation numerically using finite difference methods with dirichlet or neumann conditions. The deflection of a square plate is modeled with an elliptic PDE. ∂x4∂4z​+2∂x2∂y2∂4z​+∂y4∂4z​=Dq​ The area load is q=33.6⋅1000N with D=12(1−σ2)EΔz3​. E is the modulus of elasticity ( E=2⋅1011Pa), sigma is Poisson's ratio (σ=0.3), and Δz is the plate thickness ( Δz=10−2m) To solve this equation, we use the same trick we often used for higher order ODEs. Define a new variable, u. This allows us to solve for u, and then z. u=∂x2∂2z​+∂y2∂2z​∂x2∂2u​+∂y2∂2u​=Dq​​ Part a) Solve ∂x2∂2u​+∂y2∂2u​=Dq​ with Δx=Δy=0.5. Assume the boundary conditions are zero on each side of the square (fixed). Use 9 nodes, where your solution "usol" is defined as a column of nodes, u11, u21, u31, u12, u22, u32, u13, u23, u33. Part b) Solve ∂x2∂2z​+∂y2∂2z​=u with the same spacing. Assume the boundary conditions are zero on each side of the square. Use 9 nodes, this time with your solution to part a as your B vector. Name this output "zsol". Part c) Use the reshape() command to convert your node values in z to a 3×3 matrix of nodes. Add zeros on the top, bottom, and sides to represent the full plate. Your final matrix should be named "Zm" with dimensions 5×5. Use contour() to plot your solution and check, it should look like the one shown below.